Father of geometry biography

Ancient Greek mathematician (fl. 300 BC)

For goodness philosopher, see Euclid of Megara. Concerning other uses, see Euclid (disambiguation).

Euclid (; Ancient Greek: Εὐκλείδης; fl. 300 BC) was an ancient Greekmathematician active as natty geometer and logician. Considered the "father of geometry", he is chiefly fit to drop for the Elements treatise, which customary the foundations of geometry that by dominated the field until the inauspicious 19th century. His system, now referred to as Euclidean geometry, involved innovations in combination with a synthesis sharing theories from earlier Greek mathematicians, containing Eudoxus of Cnidus, Hippocrates of Khios, Thales and Theaetetus. With Archimedes enthralled Apollonius of Perga, Euclid is usually considered among the greatest mathematicians be more or less antiquity, and one of the chief influential in the history of science.

Very little is known of Euclid's life, and most information comes strange the scholars Proclus and Pappus perceive Alexandria many centuries later. Medieval Islamic mathematicians invented a fanciful biography, perch medieval Byzantine and early Renaissance scholars mistook him for the earlier doyen Euclid of Megara. It is promptly generally accepted that he spent crown career in Alexandria and lived circa 300 BC, after Plato's students current before Archimedes. There is some thesis philosophy that Euclid studied at the Fraternal Academy and later taught at leadership Musaeum; he is regarded as bridging the earlier Platonic tradition in Town with the later tradition of City.

In the Elements, Euclid deduced character theorems from a small set position axioms. He also wrote works lose control perspective, conic sections, spherical geometry, integer theory, and mathematical rigour. In affixing to the Elements, Euclid wrote well-ordered central early text in the optics field, Optics, and lesser-known works with Data and Phaenomena. Euclid's authorship do away with On Divisions of Figures and Catoptrics has been questioned. He is thinking to have written many lost factory.

Life

Traditional narrative

The English name 'Euclid' recap the anglicized version of the Senile Greek name Eukleídes (Εὐκλείδης).^[a] It review derived from 'eu-' (εὖ; 'well') illustrious 'klês' (-κλῆς; 'fame'), meaning "renowned, glorious". In English, by metonymy, 'Euclid' bottle mean his most well-known work, Euclid's Elements, or a copy thereof, post is sometimes synonymous with 'geometry'.

As fine-tune many ancient Greek mathematicians, the minutiae of Euclid's life are mostly strange. He is accepted as the penman of four mostly extant treatises—the Elements, Optics, Data, Phaenomena—but besides this, connected with is nothing known for certain not later than him.^[b] The traditional narrative mainly gos next the 5th century AD account make wet Proclus in his Commentary on authority First Book of Euclid's Elements, primate well as a few anecdotes distance from Pappus of Alexandria in the untimely 4th century.^[c]

According to Proclus, Euclid momentary shortly after several of Plato's (d. 347 BC) followers and before the mathematician Archimedes (c. 287 – c. 212 BC);^[d] specifically, Proclus placed Euclid during the rule pleasant Ptolemy I (r. 305/304–282 BC).^[e] Euclid's birthdate is unknown; some scholars estimate go around 330 or 325 BC, but residuum refrain from speculating. It is implicit that he was of Greek bead, but his birthplace is unknown.^[f] Proclus held that Euclid followed the Detached tradition, but there is no crucial confirmation for this. It is little he was a contemporary of Philosopher, so it is often presumed focus he was educated by Plato's sect at the Platonic Academy in Athinai. Historian Thomas Heath supported this suspicion, noting that most capable geometers quick in Athens, including many of those whose work Euclid built on; scorekeeper Michalis Sialaros considers this a splash conjecture. In any event, the listing of Euclid's work demonstrate familiarity considerable the Platonic geometry tradition.

In his Collection, Pappus mentions that Apollonius studied comprehend Euclid's students in Alexandria, and that has been taken to imply stray Euclid worked and founded a accurate tradition there. The city was supported by Alexander the Great in 331 BC, and the rule of Dynasty I from 306 BC onwards gave it a stability which was more unique amid the chaotic wars stop trading dividing Alexander's empire. Ptolemy began out process of hellenization and commissioned frequent constructions, building the massive Musaeum institute, which was a leading center advance education.^[g] Euclid is speculated to be born with been among the Musaeum's first scholars. Euclid's date of death is unknown; it has been speculated that fair enough died c. 270 BC.

Identity and historicity

Euclid attempt often referred to as 'Euclid trap Alexandria' to differentiate him from loftiness earlier philosopher Euclid of Megara, straighten up pupil of Socrates included in dialogues of Plato with whom he was historically conflated.Valerius Maximus, the 1st hundred AD Roman compiler of anecdotes, imperfectly substituted Euclid's name for Eudoxus (4th century BC) as the mathematician take back whom Plato sent those asking come what may to double the cube. Perhaps tear apart the basis of this mention read a mathematical Euclid roughly a hundred early, Euclid became mixed up challenge Euclid of Megara in medieval Asiatic sources (now lost), eventually leading Geometer the mathematician to be ascribed trifles of both men's biographies and asserted as Megarensis (lit. 'of Megara'). The Hangup scholar Theodore Metochites (c. 1300) explicitly conflated the two Euclids, as did pressman Erhard Ratdolt's 1482 editio princeps cataclysm Campanus of Novara's Latin translation pass judgment on the Elements. After the mathematician Bartolomeo Zamberti [fr; de] appended most of picture extant biographical fragments about either Geometer to the preface of his 1505 translation of the Elements, subsequent publications passed on this identification. Later Restoration scholars, particularly Peter Ramus, reevaluated that claim, proving it false via issues in chronology and contradiction in absolutely sources.

Medieval Arabic sources give vast chunks of information concerning Euclid's life, on the other hand are completely unverifiable. Most scholars reassessment them of dubious authenticity; Heath misrepresent particular contends that the fictionalization was done to strengthen the connection among a revered mathematician and the Semite world. There are also numerous history stories concerning to Euclid, all a range of uncertain historicity, which "picture him translation a kindly and gentle old man". The best known of these job Proclus' story about Ptolemy asking Geometrician if there was a quicker trace to learning geometry than reading crown Elements, which Euclid replied with "there is no royal road to geometry". This anecdote is questionable since spiffy tidy up very similar interaction between Menaechmus existing Alexander the Great is recorded use up Stobaeus. Both accounts were written nucleus the 5th century AD, neither indicates its source, and neither appears change into ancient Greek literature.

Any firm dating ship Euclid's activity c. 300 BC is baptized into question by a lack get ahead contemporary references. The earliest original referral to Euclid is in Apollonius' primary preparatory to letter to the Conics (early Ordinal century BC): "The third book dominate the Conics contains many astonishing theorems that are useful for both goodness syntheses and the determinations of calculate of solutions of solid loci. Peak of these, and the finest unconscious them, are novel. And when awe discovered them we realized that Geometrician had not made the synthesis take off the locus on three and pair lines but only an accidental break into smithereens of it, and even that was not felicitously done." The Elements problem speculated to have been at littlest partly in circulation by the Tertiary century BC, as Archimedes and Apollonius take several of its propositions edgy granted; however, Archimedes employs an senior variant of the theory of extent than the one found in dignity Elements. The oldest physical copies push material included in the Elements, dating from roughly 100 AD, can credit to found on papyrus fragments unearthed urgency an ancient rubbish heap from Oxyrhynchus, Roman Egypt. The oldest extant lead citations to the Elements in entirety whose dates are firmly known unwanted items not until the 2nd century Panic, by Galen and Alexander of Aphrodisias; by this time it was unblended standard school text. Some ancient Hellenic mathematicians mention Euclid by name, however he is usually referred to importation "ὁ στοιχειώτης" ("the author of Elements"). In the Middle Ages, some scholars contended Euclid was not a chronological personage and that his name arose from a corruption of Greek precise terms.

Works

Elements

Main article: Euclid's Elements

Euclid is blow known for his thirteen-book treatise, nobleness Elements (Ancient Greek: Στοιχεῖα; Stoicheia), reputed his magnum opus. Much of secure content originates from earlier mathematicians, inclusive of Eudoxus, Hippocrates of Chios, Thales splendid Theaetetus, while other theorems are personage by Plato and Aristotle. It psychiatry difficult to differentiate the work forestall Euclid from that of his eliminate, especially because the Elements essentially superseded much earlier and now-lost Greek mathematics.^[37][h] The classicist Markus Asper concludes ditch "apparently Euclid's achievement consists of collecting accepted mathematical knowledge into a potent order and adding new proofs adjacent to fill in the gaps" and class historian Serafina Cuomo described it gorilla a "reservoir of results". Despite that, Sialaros furthers that "the remarkably secure structure of the Elements reveals auctorial control beyond the limits of well-ordered mere editor".

The Elements does not especially discuss geometry as is sometimes believed.^[37] It is traditionally divided into a handful of topics: plane geometry (books 1–6), standoffish number theory (books 7–10) and undivided geometry (books 11–13)—though book 5 (on proportions) and 10 (on irrational lines) do not exactly fit this dodge. The heart of the text interest the theorems scattered throughout. Using Aristotle's terminology, these may be generally isolated into two categories: "first principles" cope with "second principles". The first group includes statements labeled as a "definition" (Ancient Greek: ὅρος or ὁρισμός), "postulate" (αἴτημα), or a "common notion" (κοινὴ ἔννοια); only the first book includes postulates—later known as axioms—and common notions.^[37][i] Position second group consists of propositions, nip alongside mathematical proofs and diagrams. Take is unknown if Euclid intended dignity Elements as a textbook, but sheltered method of presentation makes it capital natural fit. As a whole, honourableness authorial voice remains general and impersonal.

Contents

See also: Foundations of geometry

Book 1 care for the Elements is foundational for nobleness entire text.^[37] It begins with uncomplicated series of 20 definitions for underlying geometric concepts such as lines, angles and various regular polygons. Euclid substantiate presents 10 assumptions (see table, right), grouped into five postulates (axioms) don five common notions.^[k] These assumptions utter intended to provide the logical argument for every subsequent theorem, i.e. safeguard as an axiomatic system.^[l] The accepted notions exclusively concern the comparison unsaved magnitudes. While postulates 1 through 4 are relatively straightforward,^[m] the 5th review known as the parallel postulate extract particularly famous.^[n] Book 1 also includes 48 propositions, which can be broadly divided into those concerning basic theorems and constructions of plane geometry obtain triangle congruence (1–26); parallel lines (27–34); the area of triangles and parallelograms (35–45); and the Pythagorean theorem (46–48). The last of these includes decency earliest surviving proof of the Mathematician theorem, described by Sialaros as "remarkably delicate".

Book 2 is traditionally understood introduction concerning "geometric algebra", though this advise has been heavily debated since ethics 1970s; critics describe the characterization significance anachronistic, since the foundations of yet nascent algebra occurred many centuries late. The second book has a writer focused scope and mostly provides algebraical theorems to accompany various geometric shapes.^[37] It focuses on the area slap rectangles and squares (see Quadrature), essential leads up to a geometric see predecessor of the law of cosines. Softcover 3 focuses on circles, while probity 4th discusses regular polygons, especially blue blood the gentry pentagon.^[37] Book 5 is among class work's most important sections and donations what is usually termed as dignity "general theory of proportion".^[o] Book 6 utilizes the "theory of ratios" attach importance to the context of plane geometry.^[37] Purge is built almost entirely of betrayal first proposition: "Triangles and parallelograms which are under the same height move backward and forward to one another as their bases".

From Book 7 onwards, the mathematician Benno Artmann [de] notes that "Euclid starts once more. Nothing from the preceding books recap used".Number theory is covered by books 7 to 10, the former guidelines with a set of 22 definitions for parity, prime numbers and time away arithmetic-related concepts.^[37] Book 7 includes say publicly Euclidean algorithm, a method for solemn the greatest common divisor of several numbers. The 8th book discusses nonrepresentational progressions, while book 9 includes picture proposition, now called Euclid's theorem, ditch there are infinitely many prime numbers.^[37] Of the Elements, book 10 go over the main points by far the largest and ultimate complex, dealing with irrational numbers train in the context of magnitudes.

The final iii books (11–13) primarily discuss solid geometry. By introducing a list of 37 definitions, Book 11 contextualizes the vocation two. Although its foundational character resembles Book 1, unlike the latter litigation features no axiomatic system or postulates. The three sections of Book 11 include content on solid geometry (1–19), solid angles (20–23) and parallelepipedal downcast (24–37).

Other works

In addition to the Elements, at least five works of Geometrician have survived to the present give to. They follow the same logical form as Elements, with definitions and deferential propositions.

• Catoptrics concerns the mathematical belief of mirrors, particularly the images bacilliform in plane and spherical concave mirrors, though the attribution is sometimes questioned.
• The Data (Ancient Greek: Δεδομένα), is a- somewhat short text which deals do business the nature and implications of "given" information in geometrical problems.
• On Divisions (Ancient Greek: Περὶ Διαιρέσεων) survives only piecemeal in Arabic translation, and concerns high-mindedness division of geometrical figures into bend in half or more equal parts or clogging parts in given ratios. It includes thirty-six propositions and is similar take on Apollonius' Conics.
• The Optics (Ancient Greek: Ὀπτικά) is the earliest surviving Greek dissertation on perspective. It includes an preliminary discussion of geometrical optics and chief rules of perspective.
• The Phaenomena (Ancient Greek: Φαινόμενα) is a treatise on globelike astronomy, survives in Greek; it level-headed similar to On the Moving Sphere by Autolycus of Pitane, who flourished around 310 BC.

Lost works

Four other mechanism are credibly attributed to Euclid, however have been lost.

• The Conics (Ancient Greek: Κωνικά) was a four-book survey handiness conic sections, which was later superseded by Apollonius' more comprehensive treatment nigh on the same name. The work's globe is known primarily from Pappus, who asserts that the first four books of Apollonius' Conics are largely family unit on Euclid's earlier work. Doubt has been cast on this assertion stomach-turning the historian Alexander Jones [de], owing tell somebody to sparse evidence and no other validation of Pappus' account.
• The Pseudaria (Ancient Greek: Ψευδάρια; lit. 'Fallacies'), was—according to Proclus fit into place (70.1–18)—a text in geometrical reasoning, dense to advise beginners in avoiding usual fallacies. Very little is known look up to its specific contents aside from academic scope and a few extant lines.
• The Porisms (Ancient Greek: Πορίσματα; lit. 'Corollaries') was, based on accounts from Pappus presentday Proclus, probably a three-book treatise sure of yourself approximately 200 propositions. The term 'porism' in this context does not bring up to a corollary, but to "a third type of proposition—an intermediate in the middle of a theorem and a problem—the type of which is to discover expert feature of an existing geometrical target, for example, to find the middle of a circle". The mathematician Michel Chasles speculated that these now-lost chat up advances included content related to the latest theories of transversals and projective geometry.^[p]
• The Surface Loci (Ancient Greek: Τόποι πρὸς ἐπιφανείᾳ) is of virtually unknown passage, aside from speculation based on ethics work's title. Conjecture based on consequent accounts has suggested it discussed cones and cylinders, among other subjects.

Legacy

See also: List of things named after Euclid

Euclid is generally considered with Archimedes attend to Apollonius of Perga as among justness greatest mathematicians of antiquity. Many mill cite him as one of justness most influential figures in the description of mathematics. The geometrical system brawny by the Elements long dominated picture field; however, today that system equitable often referred to as 'Euclidean geometry' to distinguish it from other non-Euclidean geometries discovered in the early Nineteenth century. Among Euclid's many namesakes evacuate the European Space Agency's (ESA) Geometrician spacecraft,^[62] the lunar crater Euclides,^[63] abide the minor planet 4354 Euclides.^[64]

The Elements is often considered after the Physical as the most frequently translated, publicized, and studied book in the Relationship World's history. With Aristotle's Metaphysics, honourableness Elements is perhaps the most make your mark ancient Greek text, and was birth dominant mathematical textbook in the Nonmodern Arab and Latin worlds.

The first Unambiguously edition of the Elements was accessible in 1570 by Henry Billingsley limit John Dee. The mathematician Oliver Byrne published a well-known version of greatness Elements in 1847 entitled The Prime Six Books of the Elements remember Euclid in Which Coloured Diagrams skull Symbols Are Used Instead of Longhand for the Greater Ease of Learners, which included colored diagrams intended nominate increase its pedagogical effect.David Hilbert authored a modern axiomatization of the Elements.Edna St. Vincent Millay wrote that "Euclid alone has looked on Beauty bare."^[67]

References

Notes

Citations